What makes a fair allocation of resources? More generally, what makes agood
allocation? Next we shall review several proposals for measuring the quality of
an allocation. The first set of proposals is based on the idea of a collective utility
function. Any given allocation yields some utilityui ∈ Rfor agenti. This utility
will usually be the result of applying agenti’s valuation function to the bundle he
receives under the allocation in question. Now we can associate an allocation with autility vector(u1, . . . ,un)∈Rn.
Definition 5. Acollective utility function (CUF)is a function f :Rn→ R. That is, a CUF returns a single collective utility value for any given utility vector (which in turn we can think of as being generated by an allocation). This
collective utility is also referred to as thesocial welfareof the corresponding allo-
cation. The following are the most important CUFs studied in the literature:
• Under theutilitarianCUF, fu(u1, . . . ,un) := Pi∈Nui, i.e., the social welfare
of an allocation is the sum of the utilities of the individual agents. This is a natural way of measuring the quality of an allocation: the higher the
average utility enjoyed by an agent, the higher the social welfare. On the
other hand, this CUF hardly qualifies asfair:an extra unit of utility awarded
to the agent currently best offcannot be distinguished from an extra unit of
utility awarded to the agent currently worst off. Note that authors simply
writing about “social welfare” are usually talking about utilitarian social welfare.
• Under theegalitarian CUF, fe(u1, . . . ,un) := min{ui | i ∈ N}, i.e., the so-
cial welfare of an allocation is taken to be the utility of the agent worst off
under that allocation. This CUF clearly does focus on fairness, but it is
less attractive in view of economic efficiency considerations. In the special
case where we are only interested in allocations where each agent receives (at most) one item, the problem of maximizing egalitarian social welfare is
also known as theSanta Claus Problem[9].
• A possible compromise is the Nash CUF, under which fn(u1. . . ,un) :=
Q
i∈Nui. Like the utilitarian CUF, this form of measuring social welfare re-
wards increases in individual utility at all levels, but more so for the weaker
agents. For instance, the vectors (1,6,5) and (4,4,4) have the same utilitar-
ian social welfare, but the latter has a higher Nash product (and intuitively is the fairer solution of the two). For the special case of just two agents, the
Nash product is discussed in more detail in the chapter onNegotiation and
Bargainingin this volume.
Any CUF gives rise to asocial welfare ordering(SWO), a transitive and com-
plete order on the space of utility vectors (in the same way as an individual utility function induces a preference relation). We can also define SWOs directly. The
most important example in this respect is theleximin ordering. For the following
definition, suppose that all utility vectors are ordered, i.e., u1 ≤ u2 ≤ · · · ≤ un.
only if there exists ak ≤nsuch thatui =vi for alli <kanduk >vk. This is a re-
finement of the idea underlying the egalitarian CUF. Under the leximin ordering,
we first try to optimize the well-being of the worst-offagent. Once our options in
this respect have been exhausted, we try to optimize the situation for the second
worst-offagent, and so forth.
SWOs have been studied using the axiomatic method in a similar manner as SWFs and SCFs. Let us briefly review three examples of axioms considered in this area.
• An SWO % is zero independent if u % v entails (u + w) % (v + w) for
anyw ∈ Rn. That is, according to this axiom, social judgments should not
change if some of the agents change their individual “zero point”. Zero in- dependence is the central axiom in a characterization of the SWOs induced by the utilitarian CUF [83, 165].
• An SWO % is independent of the common utility pace if u % v entails
(g(u1), . . . ,g(un)) % (g(v1), . . . ,g(vn)) for any increasing bijection g : R →
R. You might think ofgas a function that maps gross to net income. Then
the axiom says that we want to be able to make social judgments indepen-
dently from the details ofg(modeling the taxation laws), as long as it never
inverts the relative welfare of two individuals. The utilitarian SWO fails this axiom, but the egalitarian SWO does satisfy it.
• An SWO%satisfies thePigou-Dalton principleifu%vwheneverucan be
obtained from v by changing the individual utilities of only two agents in
such a way that their mean stays the same and their difference reduces. The
Pigou-Dalton principle plays a central role in the axiomatic characterization of the leximin ordering [165].
For an excellent introduction to the axiomatics of welfare economics, provid- ing much more detail than what is possible here, we refer to the book of Moulin [165]. Broadly speaking, the additional information carried by a valuation func- tion (on top of its ordinal content, i.e., on top of the kind of information used in voting theory), avoids some of the impossibilities encountered in the ordinal framework. For instance, if we enrich our framework with a monetary component and stipulate that each agent’s utility can be expressed as the sum of that agent’s
money and his valuation for his goods (so-called quasilinear preferences), then
we can define strategyproof mechanisms that are not dictatorial. Examples are the mechanism used in the well-known Vickrey auction and its generalizations (see
Another important fairness criterion isenvy-freeness. An allocationAof goods is envy-free if no agent would rather obtain the bundle allocated to one of the other
agents: vi(A(i))≥ vi(A(j)) for any two agentsiand j, withA(i) andA(j) denoting
the bundles of goods allocated toiand j, respectively. Note that this concept can-
not be modeled in terms of a CUF or an SWO. If we insist on allocating all goods, then an envy-free allocation will not always exist. A simple example is the case of two agents and one item that is desired by both of them: in this case, neither of the two complete allocations will be envy-free. When no envy-free allocation
is possible, then we might want to aim for an allocation that minimizes thedegree
of envy. A variety of definitions for the degree of envy of an allocation have been proposed in the literature, such as counting the number of agents experiencing some form of envy or counting the pairs of agents where the first agent envies the second [155, 55, 154].
A new application in multiagent systems may very well call for a new fairness
or efficiency criterion. However, any new idea of this kind should always be
clearly positioned with respect to the existing standard criteria, which are well motivated philosophically and deeply understood mathematically.